Problem: $ C = \left[\begin{array}{rr}-2 & -1 \\ 3 & 4\end{array}\right]$ $ F = \left[\begin{array}{rrr}4 & -1 & -1 \\ -1 & -1 & 0\end{array}\right]$ What is $ C F$ ?
Solution: Because $ C$ has dimensions $(2\times2)$ and $ F$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C F = \left[\begin{array}{rr}{-2} & {-1} \\ {3} & {4}\end{array}\right] \left[\begin{array}{rrr}{4} & \color{#DF0030}{-1} & \color{#9D38BD}{-1} \\ {-1} & \color{#DF0030}{-1} & \color{#9D38BD}{0}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{4}+{-1}\cdot{-1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{4}+{-1}\cdot{-1} & ? & ? \\ {3}\cdot{4}+{4}\cdot{-1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rrr}{-2}\cdot{4}+{-1}\cdot{-1} & {-2}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{-1} & ? \\ {3}\cdot{4}+{4}\cdot{-1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{-2}\cdot{4}+{-1}\cdot{-1} & {-2}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{-1} & {-2}\cdot\color{#9D38BD}{-1}+{-1}\cdot\color{#9D38BD}{0} \\ {3}\cdot{4}+{4}\cdot{-1} & {3}\cdot\color{#DF0030}{-1}+{4}\cdot\color{#DF0030}{-1} & {3}\cdot\color{#9D38BD}{-1}+{4}\cdot\color{#9D38BD}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-7 & 3 & 2 \\ 8 & -7 & -3\end{array}\right] $